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  "body": "\n## Definition\n\nAn *exact couple* consists of two [[abelian groups|071DDFVG4J]] $D$ and $E$ and three homomorphisms\n$$i \\colon D \\to D, \\qquad j \\colon D \\to E, \\qquad k \\colon E \\to D,$$\nsuch that the triangle is [[exact|CBDM280TDF]] at each vertex:\n\n1. $\\ker j = \\im i$;\n2. $\\ker k = \\im j$;\n3. $\\ker i = \\im k$.\n\n## Remarks\n\nThe map $d = j \\circ k \\colon E \\to E$ satisfies $d \\circ d = j (k j) k = 0$, by clause 2.\nThe *derived couple* has $D' = i(D)$ and $E' = \\ker d / \\im d$, with $i'$ the restriction of $i$, $j'(i(x)) = [j(x)]$ and $k'([e]) = k(e)$; it is again an exact couple.\nIterating gives groups $E^r$ with differentials $d^r$ such that $E^{r+1}$ is the homology of $(E^r, d^r)$, that is, a spectral sequence.\nIn practice $D$ and $E$ are bigraded, and the bidegrees of $i, j, k$ fix the bidegree of each $d^r$.\n\nExample: let $C$ be a chain complex of free abelian groups and $p$ a prime.\nThe short exact sequence $0 \\to C \\xrightarrow{p} C \\to C / pC \\to 0$ gives a long exact sequence in homology.\nIt is an exact couple with $D = H_*(C)$, $E = H_*(C/pC)$, $i$ multiplication by $p$, $j$ reduction mod $p$ and $k$ the connecting map.\n\nNon-example: take $D = E = \\Z$ with $i = 0$, $j = \\id$ and $k = 0$.\nClause 1 holds, since $\\ker j = 0 = \\im i$, and clause 2 holds, since $\\ker k = \\Z = \\im j$.\nClause 3 fails, since $\\ker i = \\Z$ while $\\im k = 0$.\n\nExact couples were introduced by William S. Massey.\n",
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